## Low-dimensional lattices. III: Perfect forms.(English)Zbl 0655.10022

An $$n$$-dimensional positive-definite quadratic form $$f$$ is perfect if its $$n(n+1)/2$$ coefficients are uniquely determined by the system of equations $$f(v^{(i)})=\min (f)$$, where $$\pm v^{(1)},\dots,\pm v^{(s)}$$ are the minimal vectors. This notion arises in the lattice sphere-packing problem (extreme forms are perfect). As the authors say, “progress in the study of perfect lattices has been steady but slow”. After the problem for $$n\leq 5$$ had been solved in the last century, the enumeration for $$n=6$$ was completed by E. S. Barnes [Acta Arith. 5, 57–79 and 205–222 (1959; Zbl 0083.04201 and Zbl 0089.02706)], but the case $$n=7$$ is still not fully solved.
The first part of the paper contains an elegant proof that the $$A_ 1$$, $$A_ 2$$, $$A_ 3$$, $$A_ 4$$ and $$D_ 4$$ root lattices give the only perfect forms for $$n\leq 4$$. If $$L$$ is a perfect lattice, the obvious condition $$s\geq n(n+1)/2$$ and general properties of minimal vectors under congruence mod 2 immediately lead to a sublattice of $$L$$ which is the desired root lattice; knowledge of its covering radius then shows that this must be all of $$L$$. In this way the authors give also an alternative approach to the maximum density of a lattice sphere-packing for $$n\leq 4$$ (modulo the general fact that a maximum is attained).
The main part of the paper describes the perfect forms for $$n=5,6$$ and, above all, the 33 known perfect forms for $$n=7$$ [cf. K. C. Stacey, J. Lond. Math. Soc. (2) 10, 97–104 (1975; Zbl 0297.10012) and J. Aust. Math. Soc., Ser. A 22, 144–164 (1976; Zbl 0332.10014)]. This is done in a more systematic and complete way than in the existing literature. In particular, 30 of those 33 forms are found to be extreme. The authors have made their calculations as a step towards the definite enumeration of the perfect seven-dimensional forms which is expected to confirm the known list.
Reviewer: H.G.Quebbemann

### MSC:

 11E16 General binary quadratic forms 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 11H06 Lattices and convex bodies (number-theoretic aspects) 11H31 Lattice packing and covering (number-theoretic aspects)
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